Non-homogeneous Kernel Research Articles

Parameter Conditions for Boundedness of Two Integral Operators in Weighted Lebsgue Space and Calculation of Operator Norms

Firstly, Hilbert-type integral inequalities with best constant factors are established for two non-homogeneous kernels. Then, by utilizing the relationship between the Hilbert-type inequality and the integral operator of same kernel, the parameter conditions for two integral operators with non-homogeneous kernels in weighted Lebesgue space to be bounded and the formula for calculating the operator norm are obtained.

Equivalent conditions of a reverse Hilbert - type integral inequality in the whole plane and applications

In the present paper we establish a few equivalent conditions of a reverse Hilbert-type integral inequality with a general non-homogeneous kernel in the whole plane. In the form of applications, we deduce a few equivalent conditions of a reverse Hilbert-type integral inequality with a general homogeneous kernel in the whole plane. We also consider some particular cases and examples.

Equivalent Statements of Two Multidimensional Hilbert-Type Integral Inequalities with Parameters

By means of the weight functions, the idea of introduced parameters and the transfer formulas, two multidimensional Hilbert-type integral inequalities with the general nonhomogeneous kernel as H(||x||αλ1||y||βλ2)(λ1,λ2≠0) are given, which are some extensions of the Hilbert-type integral inequalities in the two-dimensional case. Some equivalent conditions of the best value and several parameters related to the new inequalities are provided. Two corollaries regarding the kernel, represented as kλ(||x||αλ1,||y||βλ2)(λ1,λ2≠0), are given, and a few new inequalities for the particular parameters are obtained.

A new reverse half-discrete Mulholland-type inequality with a nonhomogeneous kernel

In this paper, a new reverse half-discrete Mulholland-type inequality with the nonhomogeneous kernel of the form h(v(x)ln n) and the best possible constant factor is obtained by using the weight functions and the technique of real analysis. The equivalent reverses are considered. As corollaries, we deduce some new equivalent reverse inequalities with the homogeneous kernel of the form k_{lambda }(v(x),ln n). A few particular cases are provided. Our new reverse half-discrete Mulholland-type inequality which has a nonhomogeneous kernel is more general than in the previous homogeneous kernel work. The harmonized integration will have more applications.

An Equivalent Form Related to a Hilbert-Type Integral Inequality

In the present paper, we establish an equivalent form related to a Hilbert-type integral inequality with a non-homogeneous kernel and a best possible constant factor. We also consider the case of homogeneous kernel as well as certain operator expressions.

Equivalent Conditions of the Reverse Hardy-Type Integral Inequalities

Hardy-type integral inequalities play a prominent role in the study of analytic inequalities, which are essential in mathematical analysis and its various applications, such as in the study of symmetry and asymmetry phenomena. In this paper, employing methods of real analysis and using weight functions, we investigate some equivalent conditions of two kinds of reverse Hardy-type integral inequalities with a particular non-homogeneous kernel. A few equivalent conditions of two kinds of reverse Hardy-type integral inequalities with a particular homogeneous kernel are deduced in the form of applications.

A Hardy–Hilbert-type integral inequality involving two multiple upper-limit functions

By means of the weight functions, the idea of introducing parameters and the technique of real analysis, a new Hardy–Hilbert-type integral inequality with the homogeneous kernel frac{1}{(x + y)^{lambda}} (lambda > 0) involving two multiple upper-limit functions is obtained. The equivalent statements of the best possible constant factor related to the beta and gamma functions are considered. As applications, the equivalent forms and the case of a nonhomogeneous kernel are deduced. Some particular inequalities and the operator expressions are provided.

A new Hilbert-type integral inequality with the general nonhomogeneous kernel and applications

A new Hilbert-type integral inequality with the general nonhomogeneous kernel and applications

Hilbert’s Double Series Theorem’s Extensions via the Mathieu Series Approach

The author’s research devoted to the Hilbert’s double series theorem and its various further extensions are the focus of a recent survey article. The sharp version of double series inequality result is extended in the case of a not exhaustively investigated non-homogeneous kernel, which mutually covers the homogeneous kernel cases as well. Particularly, novel Hilbert’s double series inequality results are presented, which include the upper bounds built exclusively with non-weighted ℓp–norms. The main mathematical tools are the integral expression of Mathieu (a,λ)-series, the Hölder inequality and a generalization of the double series theorem by Yang.

Equivalent conditions of a multiple Hilbert-Type integral inequality with the nonhomogeneous kernel

Equivalent conditions of a multiple Hilbert-Type integral inequality with the nonhomogeneous kernel

An equivalent property of a Hilbert-type integral inequality and its applications

Making use of complex analytic techniques as well as methods involving weight functions, we study a few equivalent conditions of a Hilbert-type integral inequality with nonhomogeneous kernel and parameters. In the form of applications we deduce a few equivalent conditions of a Hilbert-type integral inequality with homogeneous kernel, and we additionally consider operator expressions.

Equivalent Properties of Two Kinds of Hardy-Type Integral Inequalities

In this paper, using weight functions as well as employing various techniques from real analysis, we establish a few equivalent conditions of two kinds of Hardy-type integral inequalities with nonhomogeneous kernel. To prove our results, we also deduce a few equivalent conditions of two kinds of Hardy-type integral inequalities with a homogeneous kernel in the form of applications. We additionally consider operator expressions. Analytic inequalities of this nature and especially the techniques involved have far reaching applications in various areas in which symmetry plays a prominent role, including aspects of physics and engineering.

The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of nonhomogeneous kernels and their applications

Abstract In this paper, we establish equivalent parameter conditions for the validity of multiple integral half-discrete Hilbert-type inequalities with the nonhomogeneous kernel G ( n λ 1 ∥ x ∥ m , ρ λ 2 ) G\left({n}^{{\lambda }_{1}}\parallel x{\parallel }_{m,\rho }^{{\lambda }_{2}}\hspace{-0.16em}) ( λ 1 λ 2 > 0 {\lambda }_{1}{\lambda }_{2}\gt 0 ) and obtain best constant factors of the inequalities in specific cases. In addition, we also discuss their applications in operator theory.

Conditions for the validity of a class of optimal Hilbert type multiple integral inequalities with nonhomogeneous kernels

For the Hilbert type multiple integral inequality ∫R+n∫R+mK(∥x∥m,ρ,∥y∥n,ρ)f(x)g(y)dxdy≤M∥f∥p,α∥g∥q,β\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\int _{\\mathbb{R}_{+}^{n}} \\int _{\\mathbb{R}_{+}^{m}} K\\bigl( \\Vert x \\Vert _{m,\\rho }, \\Vert y \\Vert _{n, \\rho }\\bigr) f(x)g(y) \\,\\mathrm{d} x \\,\\mathrm{d} y \\leq M \\Vert f \\Vert _{p, \\alpha } \\Vert g \\Vert _{q, \\beta } $$\\end{document} with a nonhomogeneous kernel K(|x|_{m, rho }, |y|_{n, rho })=G(|x|^{lambda _{1}}_{m, rho }/ |y|^{lambda _{2}}_{n, rho }) (lambda _{1}lambda _{2}> 0), in this paper, by using the weight function method, necessary and sufficient conditions that parameters p, q, lambda _{1}, lambda _{2}, α, β, m, and n should satisfy to make the inequality hold for some constant M are established, and the expression formula of the best constant factor is also obtained. Finally, their applications in operator boundedness and operator norm are also considered, and the norms of several integral operators are discussed.

On a multiple Hilbert-type integral inequality involving the upper limit functions

By applying the weight functions, the idea of introducing parameters and the technique of real analysis, a new multiple Hilbert-type integral inequality involving the upper limit functions is given. The constant factor related to the gamma function is proved to be the best possible in a condition. A corollary about the case of the nonhomogeneous kernel and some particular inequalities are obtained.

A reverse Hardy\u2013Hilbert-type integral inequality involving one derivative function

In this article, by using weight functions, the idea of introducing parameters, the reverse extended Hardy–Hilbert integral inequality and the techniques of real analysis, a reverse Hardy–Hilbert-type integral inequality involving one derivative function and the beta function is obtained. The equivalent statements of the best possible constant factor related to several parameters are considered. The equivalent form, the cases of non-homogeneous kernel and some particular inequalities are also presented.

On Hardy-type integral inequalities in the whole plane related to the extended Hurwitz-zeta function

Using weight functions, we establish a few equivalent statements of two kinds of Hardy-type integral inequalities with nonhomogeneous kernel in the whole plane. The constant factors related to the extended Hurwitz-zeta function are proved to be the best possible. In the form of applications, we deduce some special cases involving homogeneous kernel. We additionally consider some particular inequalities and operator expressions.

On a new extended half-discrete Hilbert\u2019s inequality involving partial sums

By applying the weight functions, the idea of introducing parameters, and Euler–Maclaurin summation formula, a new extended half-discrete Hilbert’s inequality with the homogeneous kernel and the beta, gamma function is given. The equivalent statements of the best possible constant factor related to a few parameters are considered. As applications, a corollary about the case of the non-homogeneous kernel and some particular cases are obtained.

On a new Hilbert-type integral inequality involving the upper limit functions

By applying the weight functions and the idea of introduced parameters we give a new Hilbert-type integral inequality involving the upper limit functions and the beta and gamma functions. We consider equivalent statements of the best possible constant factor related to a few parameters. As applications, we obtain a corollary in the case of a nonhomogeneous kernel and some particular inequalities.

EQUIVALENT PROPERTY OF A MORE ACCURATE HALF-DISCRETE HILBERT'S INEQUALITY

By using the weight functions, the idea of introducing parameters, and Hermite-Hadamard's inequality, a more accurate half-discrete Hilbert's inequality with the nonhomogeneous kernel and its equivalent form are given. The equivalent statements of the best possible constant factor related to parameters, the operator expressions and some particular cases are considered. The cases of the relating homogeneous kernel are also deduced.